8. Matchings and Factors

Transcription

1 8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council, and council policy states that no senator can be the official representative for more than one committee. For example, let there be five committees 1, 2, 3, 4 and 5. Let A, C, D, E be members of committee 1; B, C, D be members of committee 2; A, B, E, F be members of committee 3; and A, F be members of committee 5. Here the executive council can be formed by members A, C, B, F and E, representing committees 1, 2, 3, 4 and 5 respectively. The problem of formation of the executive council and many related problems can be solved using the concept of matchings in graphs. 8.1 Matchings Definition: A matching in a graph is a set of independent edges. That is, a subset M of the edge set E of a graph G(V, E) is a matching if no two edges of M have a common vertex. A matching M is said to be maximal if there is no matching M strictly containing M, that is, M is maximal if it cannot be enlarged. A matching M is said to be maximum if it has the largest possible cardinality. That is, M is maximum if there is no matching M such that M > M. Consider the graph G shown in Figure 8.1. The examples of matchings in G are M 1 = {v 1 v 5, v 2 v 6, v 3 v 4, v 7 v 8 }, M 2 = {v 1 v 5, v 2 v 6, v 3 v 4, v 7 v 9 }, M 3 = {v 1 v 5, v 2 v 6, v 3 v 4, v 7 v 10 }, M 4 = {v 1 v 5, v 2 v 6, v 3 v 4 } and M 5 = {v 4 v 7, v 1 v 6, v 2 v 3 }. Clearly, M 1 is a maximum matching, whereas M 5 is maximal but not maximum. Fig. 8.1

2 Graph Theory 199 The cardinality of a maximum matching is denoted by α 1 (G) and is called the matching number of G (or the edge-independence number of G). Definition: Let M be a matching in a graph G. A vertex v in G is said to be M-saturated (or saturated by M) if there is an edge e M incident with v. A vertex which is not incident with any edge of M is said to be M-unsaturated. In other words, given a matching M in a graph G, the vertices belonging to the edges of M are M-saturated and the vertices not belonging to the edges of M are M-unsaturated. Consider the graph shown in Figure 8.2. Clearly, M = {v 1 v 2, v 3 v 7, v 4 v 5 } is a matching and the vertices v 1, v 2, v 3, v 4, v 5, v 7 are M- saturated but v 6 is M-unsaturated. Fig. 8.2 Definition: A matching M in a graph G is said to be a perfect matching if M saturates every vertex of G. In Figure 8.3(a), G 1 has a perfect matching M = {v 1 v 2, v 3 v 6, v 4 v 5 }, but G 2 has no perfect matching. Fig. 8.3(a) Definition: If G is a bipartite graph with bipartition V = V 1 V 2 and V 1 = n 1, V 2 = n 2, then a matching M of G saturating all the vertices in V 1 is called a complete matching (or V 1 matched into V 2 ). Definition: Let H be a subgraph of a graph G. An H-alternating path (cycle) in G is a path (cycle) whose edges are alternately in E(G) E(H) and E(H). If M is matching in a graph G, then an M-alternating path (cycle) in G is a path (cycle) whose edges are alternately in E(G) M and M. That is, in an M-alternating path, the edges

3 200 Matchings and Factors alternate between M-edges and non-m-edges. An M-alternating path whose end vertices are M-unsaturated is said to be an M-augmenting path. Fig. 8.3(b) Consider the graph G shown in Figure 8.3(b). An example of a matching in G is M = {v 2 v 10, v 3 v 9, v 4 v 8, v 5 v 7 }. Clearly v 10, v 2, v 3, v 9, v 8, v 4 is an M-alternating path and v 1, v 10, v 2, v 9, v 3, v 11 is an M-augmenting path. The following result due to Berge [21] characterises a maximum matching in a graph. This forms the foundation of an efficient algorithm for obtaining a maximum matching. Theorem 8.1 (Berge) A matching M of a graph G is maximum if and only if G contains no M-augmenting paths. Proof Let M be a maximum matching in a graph G. Assume P = v 1 v 2...v k is an M- augmenting path in G. Due to the alternating nature of M-augmenting path, we observe that k is even and the edges v 2 v 3, v 4 v 5,..., v k 2 v k 1 belong to M. Also the edges v 1 v 2, v 3 v 4,..., v k 1 v k do not belong to M (Figure 8.4). Now, let M 1 be the set of edges given by Fig. 8.4 M 1 = [M {v 2 v 3,..., v k 2 v k 1 }] {v 1 v 2,..., v k 1 v k }. Then M 1 is a matching and clearly M 1 contains one more edge than M. This contradicts the assumption that M is maximum. Thus G contains no M-augmenting paths. Conversely, assume that G has no M-augmenting paths. Let M be a matching such that M > M. Let H be a subgraph of G with V(H) = V(G) and E(H) be the set of edges of G that appear in exactly one of M and M. Since every vertex of G lies on at most one edge from M and at most one edge from M, therefore degree (in H) of each vertex of H is at most 2. This implies that each connected component of H is either a single vertex, or a path, or a cycle (Fig. 8.5). If a component is a cycle, then it is an even cycle, because the edges alternate between M-edges and M -edges. Since M > M, there is at least one component in H that is a path which begins and ends with edges from M. Clearly, this path is an M-

4 Graph Theory 201 augmenting path, contradicting the assumption. Thus no such matching M can exist and hence M is maximum. Fig. 8.5 Definition: The neighbourhood of a set of vertices S, denoted by N(S), is the union of the neighbourhood of the vertices of S. The following classic result due to P. Hall [101] characterises a complete matching in a bipartite graph. Theorem 8.2 (Hall) If G(V 1, V 2, E) is a bipartite graph with V 1 V 2, then G has a matching saturating every vertex of V 1, if and only if N(S) S, for every subset S V 1. Proof Let G have a complete matching M that saturates all the vertices of V 1 and let S be any subset of V 1. Then every vertex in S is matched by M into a different vertex in N(S), so that S N(S) (Fig. 8.6 (a)). Fig. 8.6

5 202 Matchings and Factors Conversely, let N(S) S for all subsets S of V 1 and let M be a maximum matching. Assume G has no complete matching. Then there exists a vertex, say v V 1, which is M-unsaturated (Fig. 8.6(b)). Let Z be the set of vertices of G that can be joined to v by M-alternating paths. Fig. 8.7 Since M is a maximum matching, by Theorem 8.1, G has no M-augmenting path. This implies that v is the only vertex of Z that is M-unsaturated. Let S = Z V 1 and T = Z V 2 (Fig. 8.7). Then every vertex of T is matched under M to some vertex of S {v} and conversely and N(S) = T. Since T = S 1, therefore N(S) = T = S 1 < S, contradicting the given assumption. Hence G has a complete matching. The aim of the above theorem was basically to obtain the conditions for the existence of a system of distinct representatives (SDR) for a collection of subsets of a given set. Before giving such conditions, we have the following definition. Definition: If A = {A i : i N} is a family of sets, then a system of distinct representatives (SDR) of A is a set of elements {a i : i N} such that a i A i for every i N and a i a j whenever i j. For example, if A 1 = {2, 8}, A 2 = {8}, A 3 = {5, 7}, A 4 = {2, 4, 8} and A 5 = {2, 4}, then the family {A 1, A 2, A 3, A 4 } has an SDR {2, 8, 7, 4}, but the family {A 1, A 2, A 4, A 5 } has no SDR. Theorem 8.3 (Hall s SDR Theorem) A family of finite nonempty sets A = {A i : 1 i r} has an SDR if and only if for every k, 1 k r, the union of any k of these sets contains at least k elements. Proof Clearly r r A i is finite, since each of the sets A 1, A 2,..., A r is finite. Let A i = i=1 i=1 {a 1, a 2,..., a n }.

6 Graph Theory 203 Construct a bipartite graph G with partite sets X = {A 1,..., A r } and Y = {a 1,..., a n }, as shown in Figure 8.8. We take an edge between A i and a j if and only if a j A i. Fig. 8.8 Clearly, A has an SDR if and only if G has a matching that saturates all the vertices of X. Now, Theorem 8.2 implies that G has such a matching if and only if S N(S) for all subsets S of X, that is, if and only if S A i proving the result. ai S Remarks 1. Hall s theorem is often referred to as Hall s marriage theorem. 2. Since Hall s theorem, there has been remarkable progress in the theory of SDR, and besides other references the reader can refer to the book of Mirsky [161]. The following result due to M. Hall [100] counts the number of complete matchings. Theorem 8.4 If G(V 1, V 2, E) is a bipartite graph with V 1 = n 1 n 2 = V 2 and satisfying Hall s condition S N(S) for all S V 1, then G has at least r(δ, n 1 ) = 1 i min(δ, n 1 ) (δ + 1 i) complete matchings, where δ = min{d(u) : u V 1 }. Proof Induct on n 1. The result is trivial for n 1 = 1. Assume the result to be true for all values of n 1 m 1 and let G be a bipartite graph satisfying the given conditions and V 1 = m. Case 1 For each nonempty proper subset S of V 1, S < N(S). Now, for a given u V 1, choose ν N(u). This v can be chosen in at least δ ways, since δ = min{d(u) : u V 1 }. Obviously, G {u, v} is a bipartite graph with V 1 = m 1, and min{d(x) : x V 1 } δ 1. Therefore by induction hypothesis, G {u, v} has at least r(δ 1, m 1) = (δ i) complete matchings. These together with uv, give at least δ[r(δ 1, m 1)] = δ Λ complete matchings in G. 1 i min(δ 1, m 1) 1 i min(δ 1,m 1) (δ i) = (δ + 1 i) = r(δ, m) 1 i min(δ, m)

7 204 Matchings and Factors Case 2 There is a non empty proper subset S of V 1 with S = N(S). Let S = s, and G 1 =< S N(S) > and G 2 = G < S N(S) >. Then the values of n 1 and δ in G 1 are s and δ, and in G 2 are m s and δ 2 = max{δ s, 1}. Therefore G 1 and G 2 are disjoint graphs with at least r(δ, s) and r(δ 2, m s) complete matchings, by induction hypothesis. Thus G has at least r(δ, s) r(δ 2, m s) = r(δ, m) complete matchings. The following result is due to Ore [175]. Theorem 8.5 If a bipartite graph G(V 1, V 2, E) satisfies S d + N(S) for all S V 1, where d is a given positive integer, then G contains a matching M with at most d vertices of V 1 being M-unsaturated. Proof Let H(V 1, U 2, E ) be the bipartite graph with U 2 = V 2 W 2, W 2 = d and E = E [V 1, W 2 ]. Then the given condition of G implies Hall s condition for H and so H has a complete matching M. Obviously, at most d of the edges of M can be in E E, proving the result. The following observation is an immediate consequence of Theorem 8.5. Corollary 8.1 For a bipartite graph G(V 1, V 2, E), with V 1 = n 1, α 1 (G) = n 1 max{ S N(S) : S V 1 } = min{ V 1 S + N(S) : S V 1 }. The following result is a consequence of Hall s theorem. Theorem 8.6 A k( 1) regular bipartite graph has a perfect matching. Proof Let G be a k regular bipartite graph with partite sets V 1 and V 2. Then E(G) is equal to the set of edges incident to the vertices of V 1 and also E(G) is equal to the set of edges incident to the vertices of V 2. Therefore, k V 1 = k V 2 = E(G) and thus V 1 = V 2. If S V 1, then N(S) V 2 and thus N(N(S)) contains S. Let E 1 and E 2 be the sets of edges of G incident respectively to S and N(S). Then E 1 E 2, E 1 = k S and E 2 k N(S). Since E 2 E 1, N(S) S. Thus by Theorem 8.2, G has a matching that saturates all the vertices of V 1, that is G has a complete matching M. Now deleting the edges of M from G gives a (k 1)-regular bipartite graph G, with V(G) = V(G ). By the same argument, G has a complete matching M. Deletion of the edges of M from G results in a (k 2)-regular bipartite graph. Repeating this process k 1 times, we arrive at a 1-regular bipartite graph G such that V(G ) = V(G). Clearly G has a perfect matching. Definition: A set C of vertices in a graph G is said to cover the edges of G if every edge of G is incident with at least one vertex of C. Such a set C is called a covering of G. Consider the graphs G 1 and G 2 shown in Figure 8.9. Clearly, {v 1, v 2, v 4, v 5 } and {v 1, v 5, v 6 } are coverings of G 1. Also, we observe that there is no covering of G 1 with fewer than three vertices. In G 2, each of {v 1, v 2, v 6 } and {v 1, v 2, v 3, v 4, v 5, v 6 } is a covering.

8 Graph Theory 205 Fig. 8.9 The number of vertices in a minimum covering of G is called the covering number of G and is denoted by β(g). Definition: An edge covering of a graph G is a subset L of E such that every vertex of G is incident to some edge of L. Clearly, an edge covering of G exists if and only if δ > 0. The cardinality of a minimum edge covering of G is denoted by β 1 (G). For example, in the wheel W 5 of Figure 8.10, the set {v 1 v 5, v 2 v 3, v 4 v 6 } is a minimum edge covering. The following results are immediate. Fig Lemma 8.1 A subset S of V is independent if and only if V S is a covering of G. Proof Clearly, S is independent if and only if no two vertices in S are adjacent in G. Thus every edge of G is incident to a vertex of V S. This is possible if and only if V S is a covering of G. Lemma 8.2 For any graph G, α + β = n. Proof If S is a maximum independent set of G, then by Lemma 8.1, V S is a covering of G. Therefore, V S = n α β. Similarly, if C is a minimum covering of G, then V C

9 206 Matchings and Factors is independent and therefore, V S = n β α. Combining the two inequalities, we obtain α + β = n. Remark Consider the graph of Figure Clearly the set E 1 = {e 3, e 4 } is independent and E E 1 = {e 1, e 2, e 5 } is not an edge covering of G. Also, E 2 = {e 1, e 3, e 4 } is an edge covering of G and E E 2 is not independent in G. These observations imply that the edge analogue of Lemma 8.1 is not true in general. Fig The following result gives the relation between α 1 and β 1 of a graph G. Theorem 8.7(a) In a graph G with n vertices and δ > 0, α 1 + β 1 = n. Proof Let M be a maximum matching in G, so that M = α 1. Let X be the set of M- unsaturated vertices in G. Since M is maximum, X is an independent set of vertices with X = n 2α 1. As δ > 0, choose one edge for each vertex in X incident with it and let F be the set of these edges chosen. Then M F is an edge covering of G. Thus, M F = M + F = α 1 + n 2 α 1 β 1, and so n α 1 + β 1. (8.7.1) Now assume that L is a minimum edge covering of G, so that L = β 1. Let H = G(L), the edge subgraph of G defined by L and let M H be a maximum matching in H. Let the set of M H -unsaturated vertices in H be denoted by X. Since L is an edge covering of G, therefore, H is a spanning subgraph of G. Thus, L M H = L M H X = n 2 M H, and therefore L + M H n. Since M H is a matching in G, so M H α 1, and hence n β 1 + α 1. (8.7.2) Combining (8.7.1) and (8.7.2), we obtain α 1 + β 1 = n. Let M be any matching of a graph G and C be any vertex covering of G. Then in order to cover each edge of M, at least one vertex of C is to be chosen. Thus, M C. In case, M is a maximum matching and C is a minimum covering of G, then M C. We have the following obvious result.

10 Graph Theory 207 Lemma 8.3 If C is any covering and M any matching of a graph G with C = M, then C is a minimum covering and M is a maximum matching. Proof Let M be a maximum matching and C be a minimum covering of G. Then M M and C C. Therefore, M M C C. As M = C we have M = M = C = C, and the proof is complete. The next result is due to Konig [135]. Theorem 8.7(b) (Konig) In a loopless bipartite graph G, the maximum number of edges in a matching of G is equal to the minimum number of vertices in an edge cover of G, that is, α 1 = β. Proof Let G be a bipartite graph with partite sets V 1 and V 2 and let M be a matching of G. Let X be the set of all M-unsaturated vertices of V 1, so that M = V 1 X (Fig. 8.12). Fig Let A be the set of those vertices of G which are connected to some vertex of X by an M-alternating path. Further, let S = A V 1 and T = A V 2. Obviously, N(S) = T and S X is matched to T so that X = S T. Clearly, C = (V 1 S) T is a covering of G, because if there is an edge e not incident to any vertex in C, then one of the end vertices of e is in S and the other in V 2 T, which contradicts the fact that N(S) = T. Now, C = V 1 S + T = V 1 X = M. Then by Lemma 8.3, M is a maximum matching and C is a minimum covering of G. The following is a new proof of Konig s theorem due to Rizzi [224]. Second proof of Theorem 8.7 (Rizzi [224]) Let G be a minimal counter-example. Then G is connected but is not a cycle, nor is a path. Therefore G has a vertex of degree at least three. Let u be such a vertex and v one of its neighbours. If α 1 (G v) < α 1 (G), then by minimality, G v has a cover W with W < α 1 (G). Thus W {v} is a cover of G with cardinality at most α 1 (G). Therefore assume there exists a maximum matching M of G having no edge incident at v. Let f be an edge of G M incident at u but not at v. Let W

11 208 Matchings and Factors be a cover of G f with W = α 1 (G). Since no edge of M is incident at v, it follows that W does not contain v. Therefore W contains u and is a cover of G. 8.2 Factors We start this section with the following definitions. Definition: A factor F of a graph G is a spanning subgraph of G. A factor F is said to be an F factor if the components of F are some graphs in a given collection F = {H 1, H 2,..., H k } of subgraphs of G. If F contains a single graph H, then F is called an H-factor. Consider the graph G shown in Figure Clearly G 1 is a spanning subgraph whose components belong to the set F = {H 1, H 2, H 3 } and therefore is an F-factor. Each of the components of the spanning subgraph G 2 is a graph K 3 and so G 2 is a K 3 -factor. Fig We observe that a perfect matching is a K 2 -factor, since each of the components in a perfect matching is a graph K 2. A vertex function of a graph G(V, E) is a function f : V N o (N o being the set of nonnegative integers). Definition: A spanning subgraph F of a graph G(V, E) is called an f-factor of G if for a vertex function f on V, d(v F) = f(v) for all v V. For example, consider the graph G shown in Figure Define f : V N 0 by f(v i ) = 1 or 2, for all v V in G. Then the f -factor of G is given by graph H.

12 Graph Theory 209 Figure 8.14 Further, F is said to be a partial f factor (or f -matching) if d(v F) f(v), for all v V and F is said to be a covering f -factor of G if d(v F) f(v), for all v V. Some of the details of this can be seen in Graver and Jurkat [88]. We note that an f -factor exists only if v V f(v) is even. Definition: A k-factor of a graph G is a factor of G that is k-regular. Clearly, a 1-factor is a perfect matching and exists only for graphs with an even number of vertices. A 2-factor of G is a factor of G that is a disjoint union of cycles of G and connected 2-factor is a Hamiltonian cycle. Definition: If the edge set of a graph G is partitioned by the edge sets of a set of factors F 1, F 2,..., F q of G, then these factors are said to constitute a factorisation or decomposition of G and we write G = F 1 F 2... F q. If each factor F i in the factorisation is isomorphic to the same spanning subgraph F of G, then the factorisation is called an isomorphic factorisation or an F-decomposition of G. The factorisation is called an f-factorisation or k-factorisation according as each factor f i of a factorisation is an f -factor or k-factor. A graph G is f -factorable or k-factorable if G has respectively f -factorisation or k-factorisation. Example In Figure 8.15, the three distinct 1-factors of G 1 are < e 1, e 6 >, < e 2, e 4 > and < e 3, e 5 >. The two distinct 2-factors of G 2 are < e 1, e 2, e 3, e 4, e 5 > and < e 6, e 7, e 8, e 9, e 10 >. Fig. 8.15

13 210 Matchings and Factors Definition: A component of a graph is odd or even according to whether it has an odd or even number of vertices. The number of odd components of a graph G is denoted by k 0 (G). The following result of Tutte [249] characterises the graphs with a 1-factor. Several proofs of this result exist in the literature and the proof given here is due to Lovasz [150]. Theorem 8.8 (Tutte) A graph G has a 1-factor if and only if k 0 (G S) S, for all S V. (8.8.1) Proof Let G be a graph having 1-factor M. Let S be an arbitrary subset of V and let O 1, O 2,..., O k be the odd components of G S. For each i, the vertices in O i can be adjacent only to other vertices in O i and to vertices in S. Since G has a 1-factor (perfect matching), at least one vertex from each O i is to be matched with a different vertex in S. Thus the number of vertices in S is at least k, the number of odd components. Hence S k, so that k 0 (G S) S (Fig. 8.16). Fig Conversely, assume that the condition (8.8.1) holds. Let G have no 1-factor. Join pairs of non-adjacent vertices of G and continue this process till we get a maximal super graph G of G having no 1-factor. Since joining two odd components by an edge results in an even component, therefore k 0 (G S) k 0 (G S). (8.8.2) Thus for G also, we have k 0 (G S) S. When S = φ in (8.8.1), we observe that k 0 (G) = 0 so that k 0 (G ) = 0. Therefore, V(G ) = V(G) = n is even. Also, for every pair of non-adjacent vertices u and v of G, G + uv has a 1-factor and any such 1-factor definitely contains the edge uv. Now, let K be the set of those vertices of G which are having degree n 1. Clearly, K V, since otherwise G = K n has a perfect matching. Claim that each component of G K is complete. Assume, on the contrary, that in G K, there exists a component G 1 which is not complete. Then in G 1, there are vertices

14 Graph Theory 211 x, y and z such that xy, yz E(G ) and xy / E(G ). Also, y V(G 1 ) so that d(y G ) < n 1 and thus there exists a vertex w of G with yw / E(G ). Evidently, w / K (Fig. 8.17). Fig Now, by the choice of G, both G + xz and G + yw have 1-factors, say M 1 and M 2 respectively. Clearly, xz M 1 and yw M 2. Let H be the subgraph of G + {xz, yw} induced by the edges that are in M 1 and M 2 but not in both (that is, in the symmetric difference of M 1 and M 2 ). Since M 1 and M 2 are 1-factors, each vertex of G is saturated by both M 1 and M 2, and H is a disjoint union of even cycles in which the edges alternate between M 1 and M 2. We have two cases to consider. Case 1 Let xz and yw belong to different components of H, as shown in Figure 8.18 (a). If yw belongs to the cycle C, then the edge of M 1 in C together with the edges of M 2 not belonging to C form a 1-factor of G, contradicting the choice of G. Fig Case 2 Let xz and yw belong to the same component C of H. Since each component of H is a cycle, C is a cycle (Fig (b)). By symmetry of x and z, it can be assumed that the vertices x, y, w and z appear in that order on C. Then the edges of M 1 belonging to the

15 212 Matchings and Factors yw...z section of C, together with the edge yz, and the edges of M 2 not in the yw... z section of C form a 1-factor of G, again contradicting the choice of G. Thus each component of G K is complete. By conditions (8.8.2), k 0 (G K) K. Therefore one vertex of each of the odd components of G K is matched to a vertex of K. This is because each vertex of K is adjacent to every other vertex of G. Also the remaining vertices in each of the odd and even components of G K can be matched among themselves (Fig. 8.19). The total number of vertices thus matched is even. As V(G ) is even, the remaining vertices of K can be matched among themselves, thus giving a 1-factor of G, which is a contradiction. Fig Remark Among the various proofs of Theorem 8.8, one proof due to Anderson [4] uses Hall s theorem. Tutte s characterisation of graphs admitting 1-factors has been used to obtain several results including those for regular graphs and many of these are reported in the review article by Akiyama and Kano [1]. We present some of these results. The first is due to Peterson [181]. Theorem 8.9 (Peterson) Every cubic graph without cut edges has a 1-factor. Proof Let G be a cubic graph without cut edges and let S V. Let O 1, O 2,..., O k be the odd components of G S and let m i be the number of edges of G having one end in V(O i ) and the other end in S. As G is cubic, therefore, and d(v) = 3 V(O i ) (8.9.1) v V(O i ) d(v) = 3 S. (8.9.2) v S Now, E(O i ) =[V(O i ), V(O i ) S] [V(O i ), S], where [A, B] denotes the set of edges having one end in A and the other end in B, A V, B V. Thus, m i = [V(O i ), S] = d(v) v V(O i ) 2 E(O i ). Since d(v) = 3 for each v and V(O i ) is odd component, m i is odd for each i. Also,

16 Graph Theory 213 since G has no cut edges, m i 3. Therefore, k 0 (G S) = k 1 3 by Theorem 8.8, G has a 1-factor. v S d(v) < S = S. Hence Remark A cubic graph with cut edges may not have a 1-factor. To see this, consider the graph G 1 of Figure 8.20 (a). If we take S = {v}, then k 0 (G S) = 3 > 1 = S and hence G 1 has no 1-factor. Also, a cubic graph with a 1-factor may have cut edges. Consider the graph G 2 of Figure 8.20 (b). Clearly, < e 1, e 2, e 3, e 4, e 5 > is a 1-factor and e 3 is a cut edge of G 2. Fig The next result is due to Cunningham [62]. Corollary 8.2 The edge set of a simple 2-edge-connected cubic graph G can be partitioned into paths of length three. Proof By Theorem 8.9, G is a union of a 1-factor and 2-factor. Orient the edges of each cycle of the 2-factor in any manner so that each cycle becomes a directed cycle. Then, if e is any edge of the 1-factor and f 1, f 2 are the two arcs of G having their tails at the end vertices of e, then {e, f 1, f 2 } form typical 3-path of the edge partition of G (Fig. 8.21). Fig. 8.21

17 214 Matchings and Factors Berge [23] obtained the following estimate for α 1 (G) of a graph G, the proof of which is given by Bollobas [29]. Theorem 8.10 For any simple graph G, α 1 (G) = (n d)/2, where d = max{k 0 (G S) S : S V }. Proof Clearly at most n d vertices can be saturated in any matching of G. In order to show that this bound is attained, we use the following construction. Let H = GVK d. Then H satisfies Tutte s condition. For if S is any subset of V(H), then k 0 (H S ) = 0 or 1 if S V(K d ). Also, if S V(K d ), then k 0 (H S ) = k 0 (G S), where S = S V(G), so that k 0 (H S ) S = k 0 (G S) S d = 0, by definition of d. Therefore, by Tutte s theorem, H has a 1-factor M. The restriction of M of G is a matching in G which does not cover at most d vertices of G. Remark Berge actually used the theory of alternating paths to prove Theorem 8.10 and then deduced the Tutte s theorem. The restatement of Theorem 8.10 is as follows. Corollary 8.3 If G is a graph with n vertices and α 1 (G) = a, then there is a set S of s vertices such that k 0 (G S) = n+s 2a. The following result is due to Babler [10]. Theorem 8.11 Every k-regular, (k 1)-edge-connected graph with even order has a 1- factor. Proof Let G be a k-regular, (k 1)-edge-connected graph with even order and let S be any subset of V. Let O 1, O 2,..., O r be the odd components of G S with order n i and size m i, and let k i edges join O i to S, for 1 i r. Then k i k 1, (8.11.1) d(v) = kn i = 2m i + k i (8.11.2) v V(O i ) and v S d(v) = k S r k i + 2 E(S). (8.11.3) 1 Now, (8.11.2) gives k i = kn i 2m i, so that if k is even, so is k i, and thus k i > k 1 implies k i k. Similarly, if k is odd, so is k i, and again k i > k 1 implies k i k. Summing this over r odd components, we obtain by using (8.11.3) rk r i=1 k i r i=1 k i + 2 E(S) k S. Therefore, k 0 (G S) = r S and by Tutte s theorem, G has a 1-factor.

18 Graph Theory Antifactor Sets Definition: A subset S V of a graph G(V, E) such that k 0 (G S) > S is called an antifactor set. An antifactor set S such that no proper subset of S is an antifactor set is called a minimal antifactor set. For example, in the graph of Figure 8.20(a), {v} is an antifactor set. Definition: A vertex v V is a claw centre if and only if at least three vertices of G adjacent to v form an independent set. Clearly, vertex v is a claw centre in the graph of Figure 8.20(a). Now assume that G is a graph of even order n and let S be an antifactor set of G. Let k 0 (G S) = k and O 1, O 2,..., O k be the odd components of G S. Since n is even, S and k have the same parity. Choose a vertex u i V(O i ), 1 i k. Then S {u 1, u 2,..., u k } is even, that is, k + S 0 (mod 2). This implies that k S (mod 2) and therefore k 0 (G S) S (mod 2). Thus we have the following result. Lemma 8.4 If S is an antifactor set of a graph G of even order, then k 0 (G S) S + 2. Tutte s theorem implies that a graph G has either a 1-factor or an antifactor set. It is necessary to mention here that the computational complexity is high in the verification of Tutte s condition for every subset S of V. In order to reduce the computations, the antifactor sets are of great significance. The following result on antifactors is due to Sumner [236]. Lemma 8.5 If S is a minimal antifactor set of a connected graph G of even order and having no 1-factor, and if k 0 (G S) = r and S = s, then i. r s+2, ii. each vertex of S is adjacent to vertices in at least r s + 1 distinct odd components of G S, iii. every vertex of S is a claw centre. Proof i. Already established (Lemma 8.4). ii. If v S is adjacent to h distinct odd components say O 1, O 2,..., O h with h r s, then G {S {v}} has the odd components O h+1, O h+2,..., O r. Then, if h is odd, k 0 (G S ) = r h s > s 1 = S, and if h is even, then {O 1 O 2... O h } {v} is also an odd component of G S so that k 0 (G S ) = r h + 1 s+1 > s 1 = S, where S = S {v}. That is, S is an antifactor set, contradicting the minimality of S. Thus, h r s+1. iii. Since any v S is adjacent to at least r s+1 3 (by (i)) distinct odd components, v is a claw centre, by definition.

19 216 Matchings and Factors Remark The above observations imply that Tutte s theorem is equivalent to the following. A graph G has a 1-factor if and only if G does not have a set S of claw centres such that k 0 (G S) > S. Obviously, this form of Tutte s theorem decreases the number of subsets S of V to be verified in Tutte s condition. An improvement of Theorem 8.11 is due to Plesnik [209]. Theorem 8.12 If G is a k-regular, (k 1)-edge-connected graph of even order and G 1 is obtained from G by removing any k 1 edges, then G 1 has a 1-factor. Proof Assume G 1 has no 1-factor. Then G 1 has an antifactor set S with k 0 (G 1 S) > S. Therefore by Lemma 8.4, k 0 (G 1 S) S + 2. (8.12.1) Let O 1, O 2,..., O r be the odd components and O r+1, O r+2,..., O r+s be the even components of G 1 S. For a component O i, let a i and m i be the number of edges respectively of E(G) E(G 1 ) and E(G 1 ) joining it to S and b i be the number of edges of E(G) E(G 1 ) joining it to the other O j s. Then the total number of edges going out of O i is a i + b i + m i. Since G is (k 1)-edge-connected, a i + b i + m i k 1. By the argument used in Theorem 8.11, this implies that a i + b i + m i k, for an odd component O i. Summing over the r odd components, we obtain t i=1 r a i + r b i + r m i rk. (8.12.2) i=1 i=1 i=1 The number of edges between the O i s, and between O i s and S, removed from G is a i + ( ) t 1 2 b i, where t = r+ s. As this is at most k 1, we have i=1 Also, 2 t a i + t b i 2(k 1). (8.12.3) i=1 i=1 t a i + t m i d(v) = k S. (8.12.4) i=1 i=1 v S Adding (8.12.3) and (8.12.4), we get 3 t a i + t b i + t m i k( S + 2) 2. (8.12.5) i=1 i=1 i=1 As the sum on the left of (8.12.2) is less than the sum on the left of (8.12.5), we have rk k( S + 2) 2 < k( S + 2), giving r < S + 2, which contradicts (8.12.1).

20 Graph Theory 217 We now have the following observation. Corollary 8.4 A (k 1)-regular simple graph on 2k vertices has a 1-factor. Proof Assume G is a (k 1)-regular simple graph with 2k vertices having no 1-factor. Then G has an antifactor set S and by Lemma 8.4, k 0 (G S) S +2. Therefore, S +( S + 2) 2k and so S k 1. Let S = k r. Then r 1. For if r = 1, then S = k 1 and therefore k 0 (G S) = k+ 1. Thus each odd component of G S is a singleton and therefore each such vertex is adjacent to all the k 1 vertices of S, as G is (k 1) regular. This implies that every vertex of S is of degree at least k+1, a contradiction. Thus, S = k r, 2 r k 1. If G 1 is any component of G S and v V(G 1 ), then v can be adjacent to at most S vertices of S. Since G is (k 1)-regular, v is adjacent to at least (k 1) (k r) = r 1 vertices of G 1. So, V(G 1 ) r. Counting the vertices of S, we obtain ( S + 2)r+ S = 2k, or (k r+ 2)r+(k r) 2k. This gives (r 1)(r k) 0, violating the condition on r. The next result is due to Sumner [236]. Theorem 8.13 If G is a connected graph of even order n and claw-free (that is, contains no K 1, 3 as an induced subgraph), then G has a 1-factor. Proof If G has no 1-factor, then G contains a minimal antifactor set S of G. Also, there is an edge between S and each odd component of G S. If v S and vx, vy and vz are edges of G with x, y and z belonging to distinct odd components of G S, then a K 1, 3 is induced in G. This is not possible, by hypothesis. Since k 0 (G S) > S, there exists a vertex v of S and edges vu and vw of G, with u and w in distinct odd components of G S. Assume O u and O w to be the odd components containing u and w respectively. Then < O u O w {v} > is an odd component of G S 1, where S 1 = S {v}. Also, k 0 (G S 1 ) = k 0 (G S) 1 > S 1 = S 1 and therefore S 1 is an antifactor set of G with S 1 = S 1, a contradiction to the choice of S. So G has a 1-factor. (Clearly, by Lemma 8.4, the case S = 1 and k 0 (G S) = 2 does not exist). The following results are immediate, as each type of graph in these cases does not contain K 1, 3 as an induced subgraph. Corollary 8.5 Corollary 8.6 Corollary 8.7 Corollary 8.8 a 1-factor. Every connected even order edge graph has a 1-factor. If G is a connected graph of even order, then G 2 has a 1-factor. Every connected total graph of even order has a 1-factor. Every connected cubic graph in which every vertex lies on a triangle has Corollary 8.9 If a connected even order graph G has less than κ(g) claw centres, then G has a 1-factor.

21 218 Matchings and Factors The following result is the generalisation of Theorem Theorem 8.14 a 1-factor. If G is n-connected even order graph with no induced K 1, n+1, then G has Proof Assume G has no 1-factor. Then G has a minimal antifactor set S. Let S = s and let the odd components of G S be O 1, O 2,..., O k. If S i = {v S : v is adjacent to some vertex to O i }, then S i is a vertex cut of G and therefore S i n. Thus there are at least n edges from O i incident with n distinct vertices of S. The number of such edges as i ranges from 1 to k is at least nk and this is greater or equal to n(s+2), by Lemma 8.4. Thus at least one vertex v of S is incident with at least n+1 of these edges. But then, v is the centre of an induced K 1, n+1, contradicting the assumption. It can be observed that a graph can have more than one 1-factors and different 1-factors can either have no edge in common or have some edges in common. Depending on the nature of edges, we have the following definition. Definition: In a given graph, two 1-factors are said to be disjoint if they have no edge in common and two 1-factors are said to be distinct if they have at least one different edge. For example, in the graph of Figure 8.22, the 1-factors {e 1, e 2, e 3 } and {e 4, e 5, e 6 } are disjoint, while as the 1-factors {e 1, e 2, e 3 } and {e 1, e 7, e 5 } are distinct. Fig Consider maximal sets S which satisfy Tutte s condition in graphs admitting a 1-factor. If such a graph is connected, then definitely there exist subsets S such that k 0 (G S) = S. For such a graph will have a vertex v which is not a cut vertex, and G v being connected and of odd order, k 0 (G v) = 1 = v. The maximal subsets S for which k 0 (G S) = S holds is called a maximal factor set. A result on counting distinct 1-factors for bipartite graphs is given by Theorem 8.4. The next result as reported in Bollobas [29] gives the bounds for distinct 1-factors in a graph.. Theorem 8.15 Let G be a graph admitting a 1-factor and let S 0 be a maximal factor set of G with S 0 = m and O 1, O 2,..., O m be the odd components of G S 0. Further, let d = min { N(O i ) S o : 1 i m} and d be the minimum number of edges joining O i to S 0, 1 i m. Then the number F(G) of 1-factors of G satisfies

22 Graph Theory 219 i. F(G) r(d, m) = min(d,m) (d + 1 i) and i=1 ii. F(G) r (d, m), where r (d, m) is the minimum number of 1-factors in a bipartite multigraph with m vertices in each class, in which d is the minimum degree in the first vertex class. Proof Let S 0 be a maximal factor set of G so that k 0 (G S 0 ) = S 0. Using Hall s theorem for a complete matching in a bipartite graph, we can show that S 0 odd components of G S 0 can be paired with the vertices of S 0. The sufficient condition is that any k such odd components should be joined in G to at least k vertices in S 0. Clearly, this is satisfied here. For if some set of k odd components are joined in G only to a subset T of S 0 with h vertices, h < k, then k 0 (G T) k > h = T, contradicting the hypothesis. Therefore one vertex each of S 0 odd components can be matched to one vertex of S 0. Let O be an odd component of which a vertex a has been matched with a corresponding vertex s of S 0. Clearly, O = O a is a graph on an even number of vertices. Now we verify Tutte s condition on O. If there is a T V = V(O ) such that k 0 (O T) > T, then k 0 (O T) > T + 2. Let S = S 0 T {a}. Then k 0 (G S) = k 0 (O T)+k 0 (G S 0 ) 1 = k 0 (O T)+ S 0 1> T + S 0 +1= S, contradicting the choice of S 0. Thus Tutte s condition is satisfied for O and O has a 1-factor. If G S 0 has an even component C and a V(C), then G (S 0 {a}) has at least one more odd component than G S 0, contradicting the choice of S 0 or the Tutte condition. Thus G S 0 has no even components. Therefore we have seen that for each o i O i such that there is an s i S 0 with o i s i E, each O i o i has a 1-factor forming part of a 1-factor of G. The number of such choices of m independent edges o i s i equals the number of 1-factors in a bipartite graph H(V 1 V 2, E), where V 1 has m vertices, one corresponding to each odd component O i of G S 0 and V 2 has the m vertices of S 0. If we take the edge set E to be such that whenever s j N(O i ) S 0, we take exactly one o i s i E, then we get an ordinary bipartite graph for H in which the minimum degree in V 1 is d = min{ N(O i ) S 0 } and then (i) follows from Theorem 8.4. However, if we take E to be such that for each edge joining O i to s j we take an edge o i s j, then we get a bipartite multigraph for H in which the least degree of a vertex in V i is d as defined and then (ii) follows. The following observations can be seen immediately. Corollary 8.10 For a k-edge-connected graph with a unique 1-factor, F(G) k. Corollary 8.11 If G is a connected graph with a unique 1-factor, then G has a cut edge which belongs to a 1-factor. The details of the following results can be found in Bollobas [29]. Theorem 8.16 If δ (n) is the minimum degree of a graph of order 2n with exactly one 1-factor, then δ(n) [log2(n+1)].

23 220 Matchings and Factors Theorem 8.17 If e(n) is the maximum size of a graph of order 2n with exactly one 1-factor, then e(n) = n The f-factor Theorem The f -factor Theorem gives a set of necessary and sufficient conditions for the existence of an f-factor in a general graph. This result was proved by Tutte [250] for loopless graphs, using the theory of alternating paths, and later again by Tutte [251], using the 1-factor Theorem. First of all, we have the following definitions. Definition: A vertex function for a general graph G(V, E) is a function f : V N 0 such that 0 f(v) d(v), for every v V. For any function f : V N 0 and for any U V, we set f(u) = f(v). v U Definition: Three pair-wise disjoint (possibly empty) subsets S, T and U of the vertex set V of a general graph G, such that V = S T U, is called a decomposition D = (S, T, U) of G. The components, say C 1, C 2,..., C k, of the vertex-induced subgraph < U > of G are called the components of U. For each C i, let h i = f(c i )+q G [T, C i ], where f is a given vertex function of G, and q G [T, C i ] denotes the number of edges of G, whose one end is in T and the other end is in C i. Then C i is said to be an odd or even component of U, depending upon h i being odd or even, and the number of odd components of U is denoted by k 0 (D, f). The f-deficiency of D for the given G is defined as δ(d, f) = k 0 (D, f) f(s)+ f(t) d(t)+q[s, T]. The decomposition D is defined to be an f-barrier of G if δ(d, f) > 0. The term decomposition is used by Graver and Jurkat [88], and Tutte calls it a G-triple. These decompositions play the same role as the antifactor sets in the theory of 1-factors. Before going into the details, we first state the f -factor Theorem. Theorem 8.18 If f is a vertex function for a graph G, then either G has an f -factor or an f -barrier but not both. It can be seen that both the statement and the proof of the f -factor Theorem as in Tutte [253] definitely have similarities to those of the 1-factor Theorem. Since there are few complications in the constructions involved which make the proof given by Tutte [253] some what lengthy, we follow here the earlier proof by Tutte [251]. Tutte [250] defines an ordinary graph G corresponding to a given graph G such that G has an f -factor if and only if G has a 1-factor and shows that the 1-factor condition corresponds to the f -factor condition, and derives the f -factor Theorem as a consequence of the 1-factor Theorem. Berge [23] uses the same graph construction to derive the maximum

24 Graph Theory 221 f -matching Theorem. Since we have general graphs under consideration, we construct G with suitable changes as reported by Parthasarthy [180], and then find that the proof still holds. Let G be a general graph with the vertex function f. This defines for each vertex v i, its degree d i = d(v i G), f i = f(v i ) and k i = d i f i. For each vertex v i of G, take two subsets X i and Y i of vertices of G, where X i = d i and Y i = k i. Let the vertices of Y i be labelled y i j, 1 j k i, arbitrarily.now, label the vertices of X i in the following manner. Label the edges of G arbitrarily as 1, 2,..., m. Since X i = number of link edges l i incident with v i plus twice the number of loops l 0 i incident with v i, a vertex x i j X i is taken corresponding to each link edge j incident with v i and two vertices x i h, xi h are taken corresponding to each loop h incident at v i. As for edges, if link j joins v i and v k, then x i j xk j is made an edge. Corresponding to loop h incident with x i h the edge xi h xi h is taken. In addition, all the edges of the complete bipartite subgraph with vertex partition {X i, Y i } are taken for each i. These constitute the graph G. The induced subgraph < X i Y i > of G is called the star graph at i and is denoted by St(i). For multi graph G, it is a complete bipartite graph, but for general graphs it will contain edges in < X i > which we call loop edges. The other edges of St(i) are called star edges and the remaining edges of G are called as link edges. This construction is illustrated in Figure In the general graph G, d(v 1 )= 5, d(v 2 ) = 3 and d(v 3 ) = 2. Let f = (3, 2, 1) be the vertex function, so that f(v 1 ) = 3, f(v 2 ) = 2 and f(v 3 ) = 1. Therefore, k 1 = 2, k 2 = 1 and k 3 = 1. Here X 1 = 5, X 2 = 3 and X 3 = 2, and Y 1 = 2, Y 2 = 1 and Y 3 = 1. Fig. 8.23

25 222 Matchings and Factors Label the edges of G as 1, 2, 3, 4 and 5 as shown. Then Y 1 = {y 1 1, y1 2 }, Y 2 = {y 2 1 } and Y 3 = {y 3 1 }, and we have X 1 = {x 1 1, x1 1, x 1 2, x1 3, x1 4 }, X 2 = {x 2 2, x2 3, x2 5 } and X 3 = {x 3 4, x3 5 }. The vertex set of G is X i Y i, 1 i 3. The edges of G are x 1 1 x1 1, x 1 2 x2 2, x1 3 x2 3, x1 4 x3 4, x2 5 x3 5 (shown by dotted lines) and the star edges (shown by bold lines). The dotted lines in G form an f -factor with f = (3, 2, 1). The following result gives a relation between f -factors of G and the 1-factors of G. Theorem 8.18 A general graph G has an f -factor if and only if G has a 1-factor. Proof First, assume G has an f -factor F with link edges L and loop edges L 0. Let F have g i link edges and fi o loop edges at the vertex v i. Then f i = g i + 2 fi 0 X i = d i = l i + 2li 0 and Y i = k i = d i f i = (l i +2li 0) (g i+2 fi 0) = (l i g i )+2(2li 0 f i 0), where l i is the number of link edges incident with v i and li 0 is the number of loops incident with v i. Let L and (L 0 ) be the set of corresponding set of link and loop edges in G. The matching consisting of these edges, keeps unsaturated all the vertices of Y i and a set of l i g i + 2(li o f i o) = k i vertices of X i, for 1 i n. Now match the vertices of Y i with these unsaturated vertices of X i using the star edges St(i) and let S be this set of edges. Clearly L U(L 0 ) S is a 1-factor in G. Conversely, assume G has a 1-factor H. As the vertices in Y i are matched only to vertices in X i, there are exactly d i (d i f i ) = f i = g i + 2 fi 0 vertices of X i which are not matched by H to vertices in Y i. Clearly, 2 fi 0 vertices from these unmatched vertices of X i correspond to loop edges of < X i > and thus are matched in pairs amongst themselves. The remaining g i vertices are matched by the link edges of H to X j vertices of other star graphs St(i) of G. If F is the set of edges of G corresponding to the loop edges and link edges, then d (v i < F >) = g i + 2 fi 0. Thus f is an f -factor of G. The following definition will be required for the graph G constructed as above. Definition: A subset W of the vertex set V of the graph G is said to be simple if the following conditions get satisfied. i. (X i W) ϕ implies X i W, ii. (Y i W) ϕ implies X i W and iii. At most one of X i and Y i is a subset of W, implies that if Y i = ϕ, then Y i W, that is X i W = ϕ. Now assume W be a simple set of V, S be the set of vertices v i of G, for each X i W and T be the set of vertices v j of G for each Y j W. We observe that S T = ϕ, and S T (V S T) is a decomposition D of V. Denoting by d(g, S) = k 0 (G S) S, for any subset S of the vertex set V of a graph, Theorem 8.8 can be restated in the following form. Theorem 8.19 d(g, S) > 0. A simple graph G has a 1-factor if and only if it has no subset S V with

26 Graph Theory 223 Theorem 8.20 If W is a simple subset of V and D is the corresponding decomposition of V, then d(g, W) = δ(d, f). Proof Since W is a simple subset of V and D is the corresponding decomposition of V, W = X j + Y j = d j + (d j f j ) = d(s)+d(t) f(t). (8.20.1) v j S v j T v j S v j T Thus, d(g, W) = k 0 (G W) d(s) d(t)+ f(t). We proceed to find k 0 (G W) and in doing so we consider the components K of G W. We have the following possibilities. i. The component K contains a single vertex of the form x i r. This is possible when Y i W and the edge r is a link edge whose other end is in Y j W. Thus, in this case, K is an odd component, and the number of such odd components is equal to the number of such link edges and equals q[s, T]. ii. The component K contains a single vertex of the form y i α, and this is possible when X i W. This again is an odd component and the number of such odd components is equal to i S Y i = (d i f i ) = d(s) f(s). i S iii. K contains a single edge of the form x i α x α corresponding to a loop of G at α. This is possible when Y i W. Clearly, this is an even component of G W. iv. K contains a single edge of the form x i rx j r, where r is the link edge of G joining vertices v i and v j. Clearly, this happens when Y i W and Y j W. This also is an even component. v. K contains more than one edge and we call such components large components. Clearly, they have one star graph St(i) as a subgraph. If K contains more than one such star graph as a proper subgraph, it will also contain all edges of G between such star graphs. The other edges of K are of the form x i rxr, j where r is an edge of G joining v i and v j, and St(i) K and Y j T. Let v i1, v i2,..., v ik be the vertices of G corresponding to the star graphs St(i j ), 1 j k, which are subgraphs of K. Let C be the induced subgraph < v i1, v i2,..., v ik > of G. Then C is a component of G S T and therefore the number of vertices of the large component K is given by n(k) = V(K) = V(St(i)) + q[c, T] v i C = (d i + d i f i )+q[c, T] v i C

28 Graph Theory 225 q[v, T]+ q[v, S] (as d(v) = f(v)) = δ(d, f)+d(v) q[v, U] q[v, T]+ q[v, S] = δ(d, f)+2q[v, S] > δ(d, f) > 0. This contradicts the choice of D and thus the claim is established. Now, suppose W = {X i : v i S} {Y j : v j T}. Then W is simple, as S and T are disjoint. Therefore by Theorem 8.20, d (G, W) = δ(d, f) > 0 and G has no 1-factor. Thus by Theorem 8.18, G has no f -factor. Sufficiency Assume that G has no f -factor, so that G has no 1-factor. Therefore, by Theorem 8.19, there is a subset W of V with d(g, W) > 0. We choose such a W with least cardinality and prove that W is simple. Claim 1 Y j W ϕ and Y j W. If this is not true, then there is a Y j such that Y j W ϕ, and Y j (V W) ϕ. Define Q = W (Y j W) (Fig. 8.24). Fig If X j W, then G W and G Q differ in one component only, the component star edges. If X j W, then each component of G W is also a component of G Q. In either case, k 0 (G Q) k 0 (G W) 1 and Q W 1. Therefore, d(g, Q) > 0, contradicting the choice of W. Claim 2 X i W ϕ implies Y i W. If this is not true, then there is an i such that X i W ϕ and Y i W. Let a X i W and define Q = W {a}. Then G W can have at most one component with a vertex belonging to Y i which is adjacent to a in G. Thus, k 0 (G W) k 0 (G Q)+1 and Q = W 1. Therefore, d(g, Q) d(g, W), contradicting the choice of W.

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